Very recently, I was shocked when Urvashi, a class 3 student on her arrival from school declared: I hate mathematics. This set me thinking and I set my self on the trail of discovering the fault lines, if any, which might have escaped the observation of the experts- although in my twilight years it is too difficult to concentrate on important matters. My stamina refuses to wear armour and bear the arms now! But what has to be done should be done especially when the matter concerns the education of the generation next.
So I browsed through the course material to which the impressionable minds are exposed to in schools. And it took me less time to understand where the shoe pinched. The books have become trendy obviously to squeeze every rupee from the pockets of the parents. But the competition among the authors to make simple concepts difficult through scholastic presentation and conceptual jargon is very clear. No wonder the parents are forced to supplement drilling mathematical sense unto young children at post school tuition shops.
Mathematics, as I understand it is mostly about counting in situations which range from real and rational to irrational and imaginary. Soon after the Big – Bang, nature by virtue of its being, started counting (of course weighing too) atoms, molecules, protons, neutrons, ions et el. A visual vector as soon as it is released, starts logging 3×108 m/sec. the human heart counts 80 beats per minute as soon as this involuntary muscle pump starts functioning. The atmosphere like some primitive neural arrangement weighs the baggage of clouds while keeping track of two variables, viz pressure and temperature. The commerce of physical world is engaged ab initio, in counting in one form or the other.
Ever since Adam and Eve counted their eggs, humans have not only been counting their chicken but all things which are of some value to them. As the linear counting became time consuming, new methods like addition and multiplication were invented and introduced. This was followed by the invention of Logarithms and finally the binary counting.
Mathematics is not a science. It is an art like say (English) language. Both are accessories to understanding the physical world. Arts flourish through practice. The greatest flaw that is evident in present educational system is that it is short on practice but long on pedantic, retailed as conceptual understanding. No wonder, even though conversant with the fundamental math operations, the students continue making mistakes while solving simple addition and subtraction questions.
The children are great learners. They pick up their mother tongues simply by listening and mimicking. No grammarian can teach a child his/her mother tongue. Unless teachers and students built up a bond no culture of learning can be fostered. But the interest in learning gains momentum only if the students are satisfied that they are progressing. Learning makes them happy. Happiness fosters confidence which fuels enthusiasm.
So the young students take the fledgling steps learning: 1,2,3,….98,99,100. Once the smattering of first steps of counting is complete, the new age teaching places the first stumbling block and starts quizzing the students. What comes before say 29 or what comes after 37? This makes the merrily marching student to stop and think. The dilemma, whether he/she has to move forward or backwards plants the first seed of doubt in his/her mind. This is followed by asking the students to count down; an infection picked up by mathematics courtesy the rocket launching spectacles. Next comes the bouncer when they are asked to juggle with < or >. As the degree of difficulty increases, their doubts get fortified and the enthusiasm takes a beating. The tragedy is that they become aware of their weaknesses much before they discover their own strengths. How much damage is done to the overall learning faculties of the students is a subject I leave for the child psychologists to evaluate?
It is unfair to put the young students through the rigours of tests at early stages. Counting does not end at 100 or 1000. It changes tact and metamorphoses into addition and multiplication. It is my conviction that every math problem is a problem of situational counting because after solving a problem, we arrive at a number which may be real, imaginary or irrational. This leads us to the question; can we draft a pattern of numbers which can guide a student directly into the domain of multiplication and him face to face with the portal of possibilities?
The inertia gained by the traditional math curriculum at the primary level is so heavy that the idea of modernization does not cross the minds of mathematicians. No expert musters the courage to say that subtraction is a mutant which distorts the field (of numbers) between addition and multiplication. The primitive math had the immediate objective of teaching the students nitty-gritty of commercial math. That is how we, 60 years ago had to grapple with monds, seers, annas and rupees etc even up to class 8. During the last 60 years the world has changed, so has the objective of mathematics changed. Today we require students to learn the language of mathematics so as to keep pace with both physical and life sciences. We can safely postpone teaching subtraction and commercial math till the students are through with rational numbers.
This leads us to examine how a rational number is defined? The usual refrain is: a number in the form of p/q; q≠0 is a rational number. A close scrutiny reveals that it is only a photographic representation; as if p/q have been caught on camera like the alpha drone in pursuit of queen bee, in her nuptial flight, at the instant when the drone is ready to fling the treasures of nature into the abyss of the future.
Consider the number 21/7. It is not a rational number. It is a whole number-lazy, indifferent having attained the highest entropy level. On the other hand 7/21 is a rational number, restless like a hydrogen ion, ready to mate and gain entropy. But 22/7 is a hybrid, part noble part rational, hence useless.
The students in their formative years, in primary classes need to practice as much as is possible, provided the protocol is arranged, so that they find themselves in the evolutionary loop and arrive at higher levels rather than be placed on a staircase to be raised physically and mentally. I cannot recall weather Macaulay in ‘minutes of education 1835’ or those who followed him laid down the rules of engagement with mathematics in India, which are partly evident even today. But it is public knowledge that in France in 1777 AD, when during the reign of Louis XV1, eleven lac beggars were officially living in France, metric system was introduced in schools, while I had to match wits with the CGS system as late as early 60’s of the 20th century. The primitiveness of our math education is palpable as much today as it was 60 years ago. Virgin ideas of simplicity beckon us, but we are still coercing the young minds to carry the burden of history on their feeble shoulders and till those lands which have already become infertile.
I have also observed that before they grasp the language or medium of instruction, the students are subjected to the tyranny of applied math, sugar quoted as story sums. This is the single most important cause that takes the students away from mathematics. Commercial math can’t be introduced before the students have mastered to navigate in the field of numbers or before the permutation and combination of numbers gets imprinted on the motherboards of their memories. Commercial math is no rocket science. These problems (except the complex time and work problems which contain less math but more riddles of Sphinx) are problems of (mathematical) parallax. We remove the parallax and get the answer yet precious years are lost to make them market savvy.
The absence of teaching systematic sequence of concepts also makes learning mathematics boring. For example, the students in class one are made conversant with shapes even fractions, yet the square root is taught in class 8. I have before me class 8 book (NCERT publication) in which 39 pages has been used to teach this concept yet the concept eludes the students although high calibre mathematical artillery has been used. Once Socrates was asked, ‘what is justice?’ Pat came the answer: that which is due to one! I have not heard or read that which could better this definition. This is called the idiom of expression-direct from mouth to mind. In this case the brain serves as a conduit only. What is the root of a square, we may ask? It is the side of a square. This is the ultimate definition. How do we calculate it? That is the technical part’ which now a days a calculator can do accurately if long division or factorisation can’t help. In the same book at page 130 (solved example silver & gold alloy) the author betrays his/her ignorance of rational numbers, because the problem in question has been solved on three pages while the solution can be had in four lines.
We cannot make an omelette without breaking an egg. Similarly the students cannot pair with the dancing numbers unless the shell of shylock, which surrounds the stage and the dancing floors of the mathematical world, is broken. A cocktail of language spruced with numbers, served daily to the students at primary levels cannot improve their appetite for learning. After all we are dealing with growing children, who lurch along, assailed as they are, by pheromones and endorphenes-temptations and regrets. They are more interested in fairy tales and cartoons. Logic is not on their priority list. The expansion of field of their imagination far outpaces their field of logic. At primary levels, they have to practice and pick up skills. Practice makes a man perfect and who knows it better than a ripe, old person. Let the period of introspection begin.